Showing papers on "Kalman filter published in 2010"

Diffusion Strategies for Distributed Kalman Filtering and Smoothing

Abstract: We study the problem of distributed Kalman filtering and smoothing, where a set of nodes is required to estimate the state of a linear dynamic system from in a collaborative manner. Our focus is on diffusion strategies, where nodes communicate with their direct neighbors only, and the information is diffused across the network through a sequence of Kalman iterations and data-aggregation. We study the problems of Kalman filtering, fixed-lag smoothing and fixed-point smoothing, and propose diffusion algorithms to solve each one of these problems. We analyze the mean and mean-square performance of the proposed algorithms, provide expressions for their steady-state mean-square performance, and analyze the convergence of the diffusion Kalman filter recursions. Finally, we apply the proposed algorithms to the problem of estimating and tracking the position of a projectile. We compare our simulation results with the theoretical expressions, and note that the proposed approach outperforms existing techniques.

State-of-Charge Estimation for Lithium-Ion Batteries Using Neural Networks and EKF

Abstract: This paper presents a method for modeling and estimation of the state of charge (SOC) of lithium-ion (Li-Ion) batteries using neural networks (NNs) and the extended Kalman filter (EKF). The NN is trained offline using the data collected from the battery-charging process. This network finds the model needed in the state-space equations of the EKF, where the state variables are the battery terminal voltage at the previous sample and the SOC at the present sample. Furthermore, the covariance matrix for the process noise in the EKF is estimated adaptively. The proposed method is implemented on a Li-Ion battery to estimate online the actual SOC of the battery. Experimental results show a good estimation of the SOC and fast convergence of the EKF state variables.

Particle filter theory and practice with positioning applications

Abstract: The particle filter (PF) was introduced in 1993 as a numerical approximation to the nonlinear Bayesian filtering problem, and there is today a rather mature theory as well as a number of successful applications described in literature. This tutorial serves two purposes: to survey the part of the theory that is most important for applications and to survey a number of illustrative positioning applications from which conclusions relevant for the theory can be drawn. The theory part first surveys the nonlinear filtering problem and then describes the general PF algorithm in relation to classical solutions based on the extended Kalman filter (EKF) and the point mass filter (PMF). Tuning options, design alternatives, and user guidelines are described, and potential computational bottlenecks are identified and remedies suggested. Finally, the marginalized (or Rao-Blackwellized) PF is overviewed as a general framework for applying the PF to complex systems. The application part is more or less a stand-alone tutorial without equations that does not require any background knowledge in statistics or nonlinear filtering. It describes a number of related positioning applications where geographical information systems provide a nonlinear measurement and where it should be obvious that classical approaches based on Kalman filters (KFs) would have poor performance. All applications are based on real data and several of them come from real-time implementations. This part also provides complete code examples.

Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations

Abstract: In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Ito-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters.

Indoor pedestrian navigation using an INS/EKF framework for yaw drift reduction and a foot-mounted IMU

Abstract: The estimation of the position of a person in a building is a must for creating Intelligent Spaces. State-of-the-art Local Positioning Systems (LPS) require a complex sensornetwork infrastructure to locate with enough accuracy and coverage. Alternatively, Inertial Measuring Units (IMU) can be used to estimate the movement of a person; a methodology that is called Pedestrian Dead-Reckoning (PDR). In this paper, we describe and implement a Kalman-based framework, called INS-EKF-ZUPT (IEZ), to estimate the position and attitude of a person while walking. IEZ makes use of an Extended Kalman filter (EKF), an INS mechanization algorithm, a Zero Velocity Update (ZUPT) methodology, as well as, a stance detection algorithm. As the IEZ methodology is not able to estimate the heading and its drift (non-observable variables), then several methods are used for heading drift reduction: ZARU, HDR and Compass. The main contribution of the paper is the integration of the heading drift reduction algorithms into a Kalman-based IEZ platform, which represents an extended PDR methodology (IEZ+) valid for operation in indoor spaces with local magnetic disturbances. The IEZ+ PDR methodology was tested in several simulated and real indoor scenarios with a low-performance IMU mounted on the foot. The positioning errors were about 1% of the total travelled distance, which are good figures if compared with other works using IMUs of higher performance.

Visual odometry based on stereo image sequences with RANSAC-based outlier rejection scheme

Abstract: A common prerequisite for many vision-based driver assistance systems is the knowledge of the vehicle's own movement. In this paper we propose a novel approach for estimating the egomotion of the vehicle from a sequence of stereo images. Our method is directly based on the trifocal geometry between image triples, thus no time expensive recovery of the 3-dimensional scene structure is needed. The only assumption we make is a known camera geometry, where the calibration may also vary over time. We employ an Iterated Sigma Point Kalman Filter in combination with a RANSAC-based outlier rejection scheme which yields robust frame-to-frame motion estimation even in dynamic environments. A high-accuracy inertial navigation system is used to evaluate our results on challenging real-world video sequences. Experiments show that our approach is clearly superior compared to other filtering techniques in terms of both, accuracy and run-time.

Statistical Analysis of Time Series

Abstract: Chapter 31 contains formulas relevant for time series analysis: 31.1. Predictions in Time Series, 31.2. Decomposition of (Economic) Time Series, 31.3. Estimation of Correlation and Spectral Characteristics, 31.4. Linear Time Series, 31.5 Nonlinear and Financial Time Series, 31.6 Multivariate Time Series, 31.7. Kalman Filter.

False data injection attacks against state estimation in wireless sensor networks

Abstract: In this paper we study the effect of false data injection attacks on state estimation carried over a sensor network monitoring a discrete-time linear time-invariant Gaussian system. The steady state Kalman filter is used to perform state estimation while a failure detector is employed to detect anomalies in the system. An attacker wishes to compromise the integrity of the state estimator by hijacking a subset of sensors and sending altered readings. In order to inject fake sensor measurements without being detected the attacker will need to carefully design his actions to fool the estimator as abnormal sensor measurements would result in an alarm. It is important for a designer to determine the set of all the estimation biases that an attacker can inject into the system without being detected, providing a quantitative measure of the resilience of the system to such attacks. To this end, we will provide an ellipsoidal algorithm to compute its inner and outer approximations of such set. A numerical example is presented to further illustrate the effect of false data injection attack on state estimation.

Nonlinear data assimilation in geosciences: an extremely efficient particle filter

Abstract: Almost all research fields in geosciences use numerical models and observations and combine these using data-assimilation techniques. With ever-increasing resolution and complexity, the numerical models tend to be highly nonlinear and also observations become more complicated and their relation to the models more nonlinear. Standard data-assimilation techniques like (ensemble) Kalman filters and variational methods like 4D-Var rely on linearizations and are likely to fail in one way or another. Nonlinear data-assimilation techniques are available, but are only efficient for small-dimensional problems, hampered by the so-called ‘curse of dimensionality’. Here we present a fully nonlinear particle filter that can be applied to higher dimensional problems by exploiting the freedom of the proposal density inherent in particle filtering. The method is illustrated for the three-dimensional Lorenz model using three particles and the much more complex 40-dimensional Lorenz model using 20 particles. By also applying the method to the 1000-dimensional Lorenz model, again using only 20 particles, we demonstrate the strong scale-invariance of the method, leading to the optimistic conjecture that the method is applicable to realistic geophysical problems. Copyright c � 2010 Royal

PROMO: Real-time prospective motion correction in MRI using image-based tracking.

Abstract: Artifacts caused by patient motion during scanning remain a serious problem in most MRI applications The prospective motion correction technique attempts to address this problem at its source by keeping the measurement coordinate system fixed with respect to the patient throughout the entire scan process In this study, a new image-based approach for prospective motion correction is described, which utilizes three orthogonal two-dimensional spiral navigator acquisitions, along with a flexible image-based tracking method based on the extended Kalman filter algorithm for online motion measurement The spiral navigator/extended Kalman filter framework offers the advantages of image-domain tracking within patient-specific regions-of-interest and reduced sensitivity to off-resonance-induced corruption of rigid-body motion estimates The performance of the method was tested using offline computer simulations and online in vivo head motion experiments In vivo validation results covering a broad range of staged head motions indicate a steady-state error of less than 10% of the motion magnitude, even for large compound motions that included rotations over 15 deg A preliminary in vivo application in three-dimensional inversion recovery spoiled gradient echo (IR-SPGR) and three-dimensional fast spin echo (FSE) sequences demonstrates the effectiveness of the spiral navigator/extended Kalman filter framework for correcting three-dimensional rigid-body head motion artifacts prospectively in high-resolution three-dimensional MRI scans

Autonomous Underwater Vehicle Navigation

Abstract: This paper considers the vehicle navigation problem for an autonomous underwater vehicle (AUV) with six degrees of freedom. We approach this problem using an error state formulation of the Kalman filter. Integration of the vehicle's high-rate inertial measurement unit's (IMU's) accelerometers and gyros allow time propagation while other sensors provide measurement corrections. The low-rate aiding sensors include a Doppler velocity log (DVL), an acoustic long baseline (LBL) system that provides round-trip travel times from known locations, a pressure sensor for aiding depth, and an attitude sensor. Measurements correct the filter independently as they arrive, and as such, the filter is not dependent on the arrival of any particular measurement. We propose novel tightly coupled techniques for the incorporation of the LBL and DVL measurements. In particular, the LBL correction properly accounts for the error state throughout the measurement cycle via the state transition matrix. Alternate tightly coupled approaches ignore the error state, utilizing only the navigation state to account for the physical latencies in the measurement cycle. These approaches account for neither the uncertainty of vehicle trajectory between interrogation and reply, nor the error state at interrogation. The navigation system also estimates critical sensor calibration parameters to improve performance. The result is a robust navigation system. Simulation and experimental results are provided.

Model Predictive Control Tuning Methods: A Review

Abstract: This paper provides a review of the available tuning guidelines for model predictive control, from theoretical and practical perspectives. It covers both popular dynamic matrix control and generalized predictive control implementations, along with the more general state-space representation of model predictive control and other more specialized types, such as max-plus-linear model predictive control. Additionally, a section on state estimation and Kalman filtering is included along with auto (self) tuning. Tuning methods covered range from equations derived from simulation/approximation of the process dynamics to bounds on the region of acceptable tuning parameter values.

Parameter Estimation and Model Selection in Computational Biology

Abstract: A central challenge in computational modeling of biological systems is the determination of the model parameters. Typically, only a fraction of the parameters (such as kinetic rate constants) are experimentally measured, while the rest are often fitted. The fitting process is usually based on experimental time course measurements of observables, which are used to assign parameter values that minimize some measure of the error between these measurements and the corresponding model prediction. The measurements, which can come from immunoblotting assays, fluorescent markers, etc., tend to be very noisy and taken at a limited number of time points. In this work we present a new approach to the problem of parameter selection of biological models. We show how one can use a dynamic recursive estimator, known as extended Kalman filter, to arrive at estimates of the model parameters. The proposed method follows. First, we use a variation of the Kalman filter that is particularly well suited to biological applications to obtain a first guess for the unknown parameters. Secondly, we employ an a posteriori identifiability test to check the reliability of the estimates. Finally, we solve an optimization problem to refine the first guess in case it should not be accurate enough. The final estimates are guaranteed to be statistically consistent with the measurements. Furthermore, we show how the same tools can be used to discriminate among alternate models of the same biological process. We demonstrate these ideas by applying our methods to two examples, namely a model of the heat shock response in E. coli, and a model of a synthetic gene regulation system. The methods presented are quite general and may be applied to a wide class of biological systems where noisy measurements are used for parameter estimation or model selection.

Incorporation of Lagrangian measurements in freeway traffic state estimation

Abstract: Cell-phones equipped with a global positioning system (GPS) provide new opportunities for location-based services and traffic estimation. When traveling on-board vehicles, these phones can be used to accurately provide position and velocity of the vehicle as probe traffic sensors. This article presents a new technique to incorporate mobile probe measurements into highway traffic flow models, and compares it to a Kalman filtering approach. These two techniques are both used to reconstruct traffic density. The first technique modifies the Lighthill–Whitham–Richards partial differential equation (PDE) to incorporate a correction term which reduces the discrepancy between the measurements (from the probe vehicles) and the estimated state (from the model). This technique, called Newtonian relaxation, “nudges” the model to the measurements. The second technique is based on Kalman filtering and the framework of hybrid systems, which implements an observer equation into a linearized flow model. Both techniques assume the knowledge of the fundamental diagram and the conditions at both boundaries of the section of interest. The techniques are designed in a way in which does not require the knowledge of on- and off-ramp detector counts, which in practice are rarely available. The differences between both techniques are assessed in the context of the Next Generation Simulation program (NGSIM), which is used as a benchmark data set to compare both methods. They are finally tested with data from the Mobile Century experiment obtained from 100 Nokia N95 mobile phones on I-880 in California on February 8, 2008. The results are promising, showing that the proposed methods successfully incorporate the GPS data in the estimation of traffic.

Cooperative Filters and Control for Cooperative Exploration

Abstract: Autonomous mobile sensor networks are employed to measure large-scale environmental fields. Yet an optimal strategy for mission design addressing both the cooperative motion control and the cooperative sensing is still an open problem. We develop strategies for multiple sensor platforms to explore a noisy scalar field in the plane. Our method consists of three parts. First, we design provably convergent cooperative Kalman filters that apply to general cooperative exploration missions. Second, we present a novel method to determine the shape of the platform formation to minimize error in the estimates and design a cooperative formation control law to asymptotically achieve the optimal formation shape. Third, we use the cooperative filter estimates in a provably convergent motion control law that drives the center of the platform formation to move along level curves of the field. This control law can be replaced by control laws enabling other cooperative exploration motion, such as gradient climbing, without changing the cooperative filters and the cooperative formation control laws. Performance is demonstrated on simulated underwater platforms in simulated ocean fields.

1-Point RANSAC for extended Kalman filtering: Application to real-time structure from motion and visual odometry: Civera et al.: 1-Point RANSAC for EK Filtering

Abstract: Javier Civera and Oscar G. Grasa Robotics, Perception and Real-Time Group, Universidad de Zaragoza, Zaragoza 50018, Spain e-mail: jcivera@unizar.es, oscgg@unizar.es Andrew J. Davison Department of Computing, Imperial College, London SW7 2AZ, United Kingdom e-mail: ajd@doc.ic.ac.uk J. M. M. Montiel Robotics, Perception and Real-Time Group, Universidad de Zaragoza, Zaragoza 50018, Spain e-mail: josemari@unizar.es

Stochastic Stability of the Extended Kalman Filter With Intermittent Observations

Abstract: In this technical note, we analyze the error behavior of the discrete-time extended Kalman filter for nonlinear systems with intermittent observations. Modelling the arrival of the observations as a random process, we show that, given a certain regularity of the system, the estimation error remains bounded if the noise covariance and the initial estimation error are small enough. We also study the effect of different measurement models on the bounds for the error covariance matrices.

A traffic model for velocity data assimilation

Abstract: This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data

Kalman filtering and smoothing solutions to temporal Gaussian process regression models

Abstract: In this paper, we show how temporal (i.e., time-series) Gaussian process regression models in machine learning can be reformulated as linear-Gaussian state space models, which can be solved exactly with classical Kalman filtering theory. The result is an efficient non-parametric learning algorithm, whose computational complexity grows linearly with respect to number of observations. We show how the reformulation can be done for Matern family of covariance functions analytically and for squared exponential covariance function by applying spectral Taylor series approximation. Advantages of the proposed approach are illustrated with two numerical experiments.

Kalman filter-based identification for systems with randomly missing measurements in a network environment

Abstract: We consider the problem of parameter estimation and output estimation for systems in a transmission control protocol (TCP) based network environment. As a result of networked-induced time delays and packet loss, the input and output data are inevitably subject to randomly missing data. Based on the available incomplete data, we first model the input and output missing data as two separate Bernoulli processes characterised by probabilities of missing data, then a missing output estimator is designed, and finally we develop a recursive algorithm for parameter estimation by modifying the Kalman filter-based algorithm. Under the stochastic framework, convergence properties of both the parameter estimation and output estimation are established. Simulation results illustrate the effectiveness of the proposed algorithms.

Real-Time Epileptic Seizure Prediction Using AR Models and Support Vector Machines

Abstract: This paper addresses the prediction of epileptic seizures from the online analysis of EEG data. This problem is of paramount importance for the realization of monitoring/control units to be implanted on drug-resistant epileptic patients. The proposed solution relies in a novel way on autoregressive modeling of the EEG time series and combines a least-squares parameter estimator for EEG feature extraction along with a support vector machine (SVM) for binary classification between preictal/ictal and interictal states. This choice is characterized by low computational requirements compatible with a real-time implementation of the overall system. Moreover, experimental results on the Freiburg dataset exhibited correct prediction of all seizures (100 % sensitivity) and, due to a novel regularization of the SVM classifier based on the Kalman filter, also a low false alarm rate.

Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation

Abstract: This review discusses recent advances in geophysical data assimilation beyond Gaussian statistical modeling, in the fields of meteorology, oceanography, as well as atmospheric chemistry. The non-Gaussian features are stressed rather than the nonlinearity of the dynamical models, although both aspects are entangled. Ideas recently proposed to deal with these non-Gaussian issues, in order to improve the state or parameter estimation, are emphasized. The general Bayesian solution to the estimation problem and the techniques to solve it are first presented, as well as the obstacles that hinder their use in high-dimensional and complex systems. Approximations to the Bayesian solution relying on Gaussian, or on second-order moment closure, have been wholly adopted in geophysical data assimilation (e.g., Kalman filters and quadratic variational solutions). Yet, nonlinear and non-Gaussian effects remain. They essentially originate in the nonlinear models and in the non-Gaussian priors. How these effects are handled within algorithms based on Gaussian assumptions is then described. Statistical tools that can diagnose them and measure deviations from Gaussianity are recalled. The following advanced techniques that seek to handle the estimation problem beyond Gaussianity are reviewed: maximum entropy filter, Gaussian anamorphosis, non-Gaussian priors, particle filter with an ensemble Kalman filter as a proposal distribution, maximum entropy on the mean, or strictly Bayesian inferences for large linear models, etc. Several ideas are illustrated with recent or original examples that possess some features of high-dimensional systems. Many of the new approaches are well understood only in special cases and have difficulties that remain to be circumvented. Some of the suggested approaches are quite promising, and sometimes already successful for moderately large though specific geophysical applications. Hints are given as to where progress might come from.

Applications of Kalman Filtering in Aerospace 1960 to the Present [Historical Perspectives]

Abstract: In the 1960s, the Kalman filter was applied to navigation for the Apollo Project, which required estimates of the trajectories of manned spacecraft going to the Moon and back. With the lives of the astronauts at stake, it was essential that the Kalman filter be proven effective and reliable before it could be used. This article is about the lead up to Kalman's work, key discoveries in the development and maturation of the filter, a sampling of its many applications in aerospace, and recognition of some who played key roles in that history.

Robust INS/GPS Sensor Fusion for UAV Localization Using SDRE Nonlinear Filtering

Abstract: The aim of this paper is to present a new INS/GPS sensor fusion scheme, based on state-dependent Riccati equation (SDRE) nonlinear filtering, for unmanned aerial vehicle (UAV) localization problem. SDRE navigation filter is proposed as an alternative to extended Kalman filter (EKF), which has been largely used in the literature. Based on optimal control theory, SDRE filter solves issues linked with EKF filter such as linearization errors, which severely decrease UAV localization performances. Stability proof of SDRE nonlinear filter is also presented and validated on a 3-D UAV flight scenario. Results obtained by SDRE navigation filter were compared to EKF navigation filter results. This comparison shows better UAV localization performance using SDRE filter. The suitability of the SDRE navigation filter over an unscented Kalman navigation filter for highly nonlinear UAV flights is also demonstrated.

Evaluating the Performances of Adaptive Kalman Filter Methods in GPS/INS Integration

Abstract: One of the most important tasks in integration of GPS/INS is to choose the realistic dynamic model covariance matrix Q and measurement noise covariance matrix R for use in the Kalman filter. The performance of the methods to estimate both of these matrices depends entirely on the minimization of dynamic and measurement update errors that lead the filter to converge. This paper evaluates the performances of adaptive Kalman filter methods with different adaptations. Innovation and residual based adaptive Kalman filters were employed for adapting R and Q. These methods were implemented in a loose GPS/INS integration system and tested using real data sets. Their performances have been evaluated and compared. Their limitations in real-life engineering applications are discussed.

Orientation Estimation Using a Quaternion-Based Indirect Kalman Filter With Adaptive Estimation of External Acceleration

Abstract: This paper is concerned with orientation estimation using inertial and magnetic sensors. A quaternion-based indirect Kalman filter structure is used. The magnetic sensor output is only used for yaw angle estimation using two-step measurement updates. External acceleration is estimated from the residual of the filter and compensated by increasing the measurement noise covariance. Using the direction information of external information, the proposed method prevents unnecessarily increasing the measurement noise covariance corresponding to the accelerometer output, which is not affected by external acceleration. Through numerical examples, the proposed method is verified.

A Kalman/Particle Filter-Based Position and Orientation Estimation Method Using a Position Sensor/Inertial Measurement Unit Hybrid System

Abstract: This paper presents a novel methodology that estimates position and orientation using one position sensor and one inertial measurement unit. The proposed method estimates orientation using a particle filter and estimates position and velocity using a Kalman filter (KF). In addition, an expert system is used to correct the angular velocity measurement errors. The experimental results show that the orientation errors using the proposed method are significantly reduced compared to the orientation errors obtained from an extended Kalman filter (EKF) approach. The improved orientation estimation using the proposed method leads to better position estimation accuracy. This paper studies the effects of the number of particles of the proposed filter and position sensor noise on the orientation accuracy. Furthermore, the experimental results show that the orientation of the proposed method converges to the correct orientation even when the initial orientation is completely unknown.

A Survey on Distributed Estimation and Control Applications Using Linear Consensus Algorithms

Abstract: In this chapter we present a popular class of distributed algorithms, known as linear consensus algorithms, which have the ability to compute the global average of local quantities. These algorithms are particularly suitable in the context of multi-agent systems and networked control systems, i.e. control systems that are physically distributed and cooperate by exchanging information through a communication network. We present the main results available in the literature about the analysis and design of linear consensus algorithms,for both synchronous and asynchronous implementations. We then show that many control, optimization and estimation problems such as least squares, sensor calibration, vehicle coordination and Kalman filtering can be cast as the computation of some sort of averages, therefore being suitable for consensus algorithms. We finally conclude by presenting very recent studies about the performance of many of these control and estimation problems, which give rise to novel metrics for the consensus algorithms. These indexes of performance are rather different from more traditional metrics like the rate of convergence and have fundamental consequences on the design of consensus algorithms.

Filtered Multitensor Tractography

Abstract: We describe a technique that uses tractography to drive the local fiber model estimation. Existing techniques use independent estimation at each voxel so there is no running knowledge of confidence in the estimated model fit. We formulate fiber tracking as recursive estimation: at each step of tracing the fiber, the current estimate is guided by those previous. To do this we perform tractography within a filter framework and use a discrete mixture of Gaussian tensors to model the signal. Starting from a seed point, each fiber is traced to its termination using an unscented Kalman filter to simultaneously fit the local model to the signal and propagate in the most consistent direction. Despite the presence of noise and uncertainty, this provides a causal estimate of the local structure at each point along the fiber. Using two- and three-fiber models we demonstrate in synthetic experiments that this approach significantly improves the angular resolution at crossings and branchings. In vivo experiments confirm the ability to trace through regions known to contain such crossing and branching while providing inherent path regularization.